Difference between revisions of "1964 AHSME Problems/Problem 20"

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==Solution==
 
==Solution==
  
For any polynomial, even a polynomial with more than one variable, the sum of all the coefficients (including the constant, which is the coefficient of <math>x^0y^0</math>) is found by setting all variables equal to <math>1</math>.  Note that we don't have to worry about whether a constant is a coefficient of an "invisible <math>x^0y^0</math>" term, because there is no such term here.
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For any polynomial, even a polynomial with more than one variable, the sum of all the coefficients (including the constant, which is the coefficient of <math>x^0y^0</math>) is found by setting all variables equal to <math>1</math>.  Note that we don't have to worry about whether a constant is a coefficient of an "invisible" <math>x^0y^0</math> term, because there is no such term here.
  
 
Setting <math>x=y=1</math> gives <math>(-1)^{18}</math>, which is equal to <math>1</math>, which is answer <math>\boxed{\textbf{(B)}}</math>.
 
Setting <math>x=y=1</math> gives <math>(-1)^{18}</math>, which is equal to <math>1</math>, which is answer <math>\boxed{\textbf{(B)}}</math>.

Latest revision as of 22:29, 23 July 2019

Problem 20

The sum of the numerical coefficients of all the terms in the expansion of $(x-2y)^{18}$ is:

$\textbf{(A)}\ 0\qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ 19\qquad \textbf{(D)}\ -1\qquad \textbf{(E)}\ -19$

Solution

For any polynomial, even a polynomial with more than one variable, the sum of all the coefficients (including the constant, which is the coefficient of $x^0y^0$) is found by setting all variables equal to $1$. Note that we don't have to worry about whether a constant is a coefficient of an "invisible" $x^0y^0$ term, because there is no such term here.

Setting $x=y=1$ gives $(-1)^{18}$, which is equal to $1$, which is answer $\boxed{\textbf{(B)}}$.

See Also

1964 AHSC (ProblemsAnswer KeyResources)
Preceded by
Problem 19
Followed by
Problem 21
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
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